-.15cmExample  Let us consider the weight \(W(z)=z^\alpha (1-z)^\beta \) , with
\(\alpha \) , \(\beta \)  scalars in \((-1, \infty )\) . Then, the scalar second order equation for \(\left\lbrace P^{\mathsf {L}}_n\right\rbrace _{n\in \mathbb {N}}\)  and \(\left\lbrace Q^{\mathsf {L}}_n\right\rbrace _{n\in \mathbb {N}}\)  (cf. for example [1]}) is given by
\( z(1-z) P_n^{\prime \prime } (z) + \big (1+\alpha - (\alpha +\beta +2)z\big ) P_n^{\prime } (z) + n(\alpha +\beta +n+1) P_n (z)&=0 , \\z(1-z) Q_n^{\prime \prime } (z) + \big (1-\alpha + (\alpha +\beta -2)z\big ) Q_n^{\prime } (z) +(n+1)(\alpha +\beta +n) Q_n (z)&=0 .\) 

